To solve the problem, we'll use the fact that the distance a spring stretches varies directly with the weight attached to it. This can be expressed with the equation:
\[ d = k \cdot w \]
where:
- \( d \) is the distance the spring stretches,
- \( w \) is the weight attached to the spring,
- \( k \) is the constant of variation.
From the problem, we know that when \( w = 80 \) pounds, the spring stretches \( d = 6 \) inches. We can use this information to find the constant \( k \):
\[ 6 = k \cdot 80 \] \[ k = \frac{6}{80} = \frac{3}{40} \]
Now that we have the value of \( k \), we can find how far the spring will stretch when \( w = 55 \) pounds:
\[ d = k \cdot w \] \[ d = \frac{3}{40} \cdot 55 \]
Calculating this value:
\[ d = \frac{3 \cdot 55}{40} = \frac{165}{40} = 4.125 \]
Now rounding to the nearest tenth, we get:
\[ d \approx 4.1 \text{ inches} \]
Thus, the spring will stretch approximately 4.1 inches with 55 pounds attached.
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